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Video Transcript

TranscriptUsing and Understanding the Discriminant

”Hello! It's that time again for your favorite quiz show
THE WALL OF FAME!””The first contestant on today's show is Theresa!
I hope you're ready to play THE WALL OF FAME!”
Before we begin today's game, let's quickly review our game rules. Theresa, you're about to see today's wall. On the wall are 5 secret doors with one equation written over each door. During the game, you will get to open 3 of those doors. For each door you pick, you will receive a prize equal to the number of solutions the equation has. If each door you pick reveals a prize, you get to come back on the show tomorrow for a chance to win an even bigger prize!

Understanding the Discriminant

The secret to this game is understanding and using the discriminant. If Theresa can figure out how to use this tool, she just might have a chance to win all the prizes and come back tomorrow! Let's take a look at today's WALL OF FAME! The equations we have today are:
Equation 1: x-squared minus 10x plus 34.
Equation 2: 3 x-squared minus 4x plus 10.
Equation 3: x-squared minus 3x plus 5.
Equation 4: x-squared plus 2 root 2x plus 2.
Equation 5: x-squared plus 6x minus 16.

With her first choice, Theresa picks door number 4. Remember, we said we could use the discriminant to quickly determine whether or not Theresa has won one of our fabulous prizes. The discriminant comes from the quadratic formula, and it's formula is b-squared minus 4ac where a, b and c refer to the coefficients and constant of a quadratic equation in standard form.

When the discriminant is positive, our equation will have 2 solutions; when the discriminant is equal to 0, our equation has 1 solution and when our discriminant is negative, our equation will have no solutions. Let's take a look at the equation Theresa has just selected. In the equation Theresa has just selected, "a" is equal to 1, "b" is equal to 2 times the square root of 2, and "c" is equal to 2.

Let's calculate the discriminant! The quantity 2 times the square root of 2 squared will be 8. 8 minus 8 will equal 0. Because the discriminant is equal to 0, this equation will have 1 solution. Theresa, you've just won 1 brand new unicorn! Doesn't she look happy?! That's about as good of a start as you can ask for!

Second Example

With her second selection, Theresa picks the fifth equation. Let's take a closer look! In this equation, "a" is equal to 1, "b" is equal to 6, and "c" is equal to negative 16. Let's calculate the discriminant! Plugging in our values, we get 6 squared minus 4 times 1 times negative 16.
Using PEMDAS, we get 36 plus 64 which equals 100. Wow! Look at that! Theresa has just won a pair of Positron and Negatron figurines! 2 prizes for an equation with 2 solutions! Theresa's doing great so far! If she can pick a door with one more prize she'll be invited back for tomorrow's show and a chance to win the Grand Prize!

And with her final selection, Theresa picks the equation on door number 2! Let's take a look at her equation! In this equation, "a" is equal to 3, "b" is equal to negative 4, and "c" is equal to 10. Let's calculate the discriminant! Plugging in our values, we get negative 4 squared minus 4 times 3 times 10. Calculating our math, we get 16 minus 120 which equals negative 104. What do you think this means? Not only for how many solutions our equation has but for Theresa's chances at coming back tomorrow?

Summary

Remember, when our discriminant is greater than 0 that means our equation will have 2 solutions. When our discriminant is equal to 0 our equation will have 1 solution. And when our discriminant is negative, our equation will have no real solutions.

Well Folks! That also means there's no prize! Although Theresa is going to be disappointed, at least she got some action figures and a new Unicorn! Although that Unicorn looks a bit suspicious... And the host's hair looks a bit suspicious too!

About this Video Lesson

Description

A quadratic equation in standard form ax² + bx + c = 0 can be solved in various ways. These include factoring, completing the square, graphing, and using the quadratic formula. Solving for the roots or solutions of a quadratic equation by using the quadratic formula is often the most convenient way. The solutions to a quadratic equation can be determined by this quadratic formula: x = [-b ± √(b²-4ac)] / 2a. We can predict the number of solutions to a quadratic equation by evaluating the discriminant given by b²-4ac or the radicand expression in the quadratic formula. The value of the discriminant determines the nature and number of solutions of a quadratic equation.